Related papers: On the maximum Zagreb indices of bipartite graphs …

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the…

For a simple graph $G$ with $n$ vertices and $m$ edges, the first Zagreb index and the second Zagreb index are defined as $M_1(G)=\sum_{v\in V}d(v)^2 $ and $M_2(G)=\sum_{uv\in E}d(u)d(v)$. In \cite{VGFAD}, it was shown that if a connected…

The first Zagreb index of a graph $G$ is the sum of the square of every vertex degree, while the second Zagreb index is the sum of the product of vertex degrees of each edge over all edges. In our work, we solve an open question about…

The second Zagreb index of a graph G is denoted by $M_2(G)=\sum_{uv\in E(G)}d(u)d(v)$. In this paper, we investigate properties of the extremal graphs with the maximum second Zagreb indices with given graphic sequences, in particular…

The Zagreb index of a hypergraph is defined as the sum of the squares of the degrees of its vertices. A connected $k$-uniform hypergraph with $n$ vertices and $m$ edges is called bicyclic if $n=m(k-1)-1$. In this paper, we determine the…

For a (molecular) graph, the first multiplicative Zagreb index $\prod_1(G) $ is the product of the square of every vertex degree, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of the products of degrees of pairs of…

Let ${\mathcal G}_n$ be the set of class of graphs of order $n$. The first Zagreb index $M_1(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_2(G)$ is equal to the sum of the products of the…

The complementary second Zagreb index of a graph $G$ is defined as $cM_2(G)=\sum_{uv\in E(G)}|(d_u(G))^2-(d_v(G))^2|$, where $d_u(G)$ denotes the degree of a vertex $u$ in $G$ and $E(G)$ represents the edge set of $G$. Let $G^*$ be a graph…

The first Zagreb index of a graph $G$ is the sum of squares of the vertex degrees in a graph and the second Zagreb index of $G$ is the sum of products of degrees of adjacent vertices in $G$. The imbalance of an edge in $G$ is the numerical…

Let $D=(V,A)$ be a digraphs without isolated vertices. The first Zagreb index of a digraph $D$ defined as a summation over all arcs, $M_1(D)=\frac{1}{2}\sum\limits_{uv\in A}(d^{+}_{u}+d^{-}_v)$, where $d^{+}_u$(resp. $d^{-}_u$) denotes the…

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper,…

For a graph $G$, the first multiplicative Zagreb index $\prod_1(G) $ is the product of squares of vertex degrees, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of products of degrees of pairs of adjacent vertices.…

Let $G = (V, E)$ be a graph. The first Zagreb index of a graph $G$ is defined as $\sum_{u \in V} d^2(u)$, where $d(u)$ is the degree of vertex $u$ in $G$. Using the P\'{o}lya-Szeg\H{o} inequality, we in this paper present the first Zagreb…

This paper presents new lower bounds for the first general Zagreb index $Z_{\alpha}(G)$ involving two, three, and four arbitrary degrees of vertices of a simple graph $G$. For the special cases $\alpha = 2$ and $\alpha = -2$, the results…

The hyper Zagreb index is a kind of extensions of Zagreb index, used for predicting physicochemical properties of organic compounds. Given a graph $G= (V(G), E(G))$, the first hyper-Zagreb index is the sum of the square of edge degree over…

Recently, a couple of degree-based topological indices, defined using a geometrical point of view of a graph edge, have attracted significant attention and being extensively investigated. Furtula and Oz [Complementary Topological Indices,…

For a molecular graph, the first multiplicative Zagreb index $\Pi_1$ is equal to the product of the square of the degree of the vertices, while the second multiplicative Zagreb index $\Pi_2$ is equal to the product of the endvertex degree…

We examine the quantity \[S(G) = \sum_{uv\in E(G)} \min(\text{deg } u, \text{deg } v)\] over sets of graphs with a fixed number of edges. The main result shows the maximum possible value of $S(G)$ is achieved by three different classes of…

The first Zagreb index $M_{1}(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_{2}(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying…

Let $G=(V,E)$ be a simple graph with $n = |V|$ vertices and $m = |E|$ edges. The first and second Zagreb indices are among the oldest and the most famous topological indices, defined as $M_1 = \sum_{i \in V} d_i^2$ and $M_2 = \sum_{(i, j)…